This file demonstrates principal component analysis (PCA) of a wiener process (Brownian motion).

Two methods are used to calculate principal components in this file:
Method 1 : Simulate Brownian motion and use eig function to calculate principal components of the matrix
Method 2 : Use analytical formula mentioned by Fredrik Akesson, John P. Lehoczky in paper Discrete eigenfunction expansion of the multidimensional brownian motion and the Ornstein-Uhlenbeck process

Following formula is used to get eigen values:


Following formula is used to get eigen vectors:


Following figure displays eigenvectors of the first 4 components:

This script demonstrates how to solve following ODE by using Galerkin method:


Based on paper http://math.fullerton.edu/mathews/n2003/galerkin/GalerkinMod/Links/GalerkinMod_lnk_5.html
After running this script, the output values are coefficients of basis functions.

Karhunen Loeve expansion is a representation of a stochastic process as sum of an infinite series of terms which are product of random variables and orthogonal functions:




As we know, a Wiener process Xt is characterized by the following properties:
1.Expected value of Xt i.e., E[Xt]=0
2.Variance is t

Here we demonstrate how KL expansion can be used for generating Wiener process paths.

A rescaled Wiener process has the following representation:


In the attached matlab program we generate IID random variables and use this formula to generate Wiener process paths. Instead of using no. of expansion terms to infinity, we restrict it to value of nk.


After doing simulations, we verify that as the no. of expansion terms increase, the simulated process converges to mean 0 and variance t. Note that t should be between 0 and 1.


Section 4.4 in this paper has nice explanation on derivation of expansion formula : http://www.cims.nyu.edu/~eve2/chap4.pdf

This code demonstrates the different type of local volatility surfaces that result by changing an input implied volatility surface.

eg., this is implied vol. surface:


This is resulting local vol. surface which is based on the above implied vol surface:


Note : Please also download LocalVolSurface to run this file

This is a very simple demonstration of doing monte carlo simulation in libor market model. It shows how to reproduce results for swap price of a swap which starts at T=20 and ends at T=20.5, as mentioned in table2 of paper Linking Caplet and Swaption Volatilities in a BGM/J Framework: Approximate Solutions

NOTE : This code will only work for the last swap and not for the other swaps. For all other swaps the calculation is more involed and has not been implemented yet.


For example, following results are obtained by running this code and using parameter CurveType="G":

Value mentioned in paper is .215

Following results are obtained using parameter CurveType="F:

Value mentioned in paper is .242




Below is formula for LMM dynamics used for simulation: